The covering lemma and $q$-analogues of extremal set theory problems

TL;DR

This paper introduces a general lemma linking the maximum size of hereditary-structured sets to their substructures, and applies it to extend forbidden subposet results from Boolean lattices to vector space subposets, including generalized cases.

Contribution

The paper presents a new covering lemma that connects hereditary properties in set systems to subspace posets, enabling transfer of extremal results between these structures.

Findings

Established a general lemma relating hereditary properties and substructures.

Extended forbidden subposet results from Boolean lattices to vector space posets.

Explored generalized forbidden subposet problems in subspace lattices.

Abstract

We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about the connection of the largest cardinalities (or weight) of structures satisfying some hereditary property and substructures satisfying the same hereditary property. We use it to show how results concerning forbidden subposet problems in the Boolean poset imply analogous results in the poset of subspaces of a finite vector space. We also study generalized forbidden subposet problems in the poset of subspaces.